Explicit Time Integration Method Research Articles

A selective mass scaling method for shear wave propagation analyses in nearly incompressible materials

SummaryThis paper describes a selective mass scaling method which is designed for the analysis of wave propagation problems in nearly incompressible materials. The incompressibility of materials leads to a high value of the compressional wave speed, which makes the time step extremely small in explicit time integration method. The proposed selective mass scaling method selects the eigenfrequencies related to volumetric deformation modes to decrease them, while it keeps the shear eigenmodes unchanged. This makes the time step no longer limited by the compressional wave speed but by the shear wave speed. A significant reduction of CPU time is obtained with a good accuracy for transient problems in small strains on free or largely prestressed media. Copyright © 2016 John Wiley & Sons, Ltd.

Minimally implicit Runge-Kutta methods for Resistive Relativistic MHD

The Relativistic Resistive Magnetohydrodynamic (RRMHD) equations are a hyperbolic system of partial differential equations used to describe the dynamics of relativistic magnetized fluids with a finite conductivity. Close to the ideal magnetohydrodynamic regime, the source term proportional to the conductivity becomes potentially stiff and cannot be handled with standard explicit time integration methods. We propose a new class of methods to deal with the stiffness fo the system, which we name Minimally Implicit Runge-Kutta methods. These methods avoid the development of numerical instabilities without increasing the computational costs in comparison with explicit methods, need no iterative extra loop in order to recover the primitive (physical) variables, the analytical inversion of the implicit operator is trivial and the several stages can actually be viewed as stages of explicit Runge-Kutta methods with an effective time-step. We test these methods with two different one-dimensional test beds in varied conductivity regimes, and show that our second-order schemes satisfy the theoretical expectations.

Nonlinear dynamic collapse analysis of semi-rigid steel frames based on the finite particle method

This paper presents a numerical procedure based on the finite particle method (FPM) for the nonlinear dynamic collapse analysis of plane steel frames with semi-rigid connections. The FPM can be used to model the plane frame with finite separated particles; in particular, geometric nonlinearity and dynamic fracture. Fictitious motion was used to consider the geometrical nonlinearity, and the explicit time integration method was used for solving the dynamic equilibrium equations. In the FPM, particles are free to separate from each other, which is advantageous in the simulation of member fracture. To simulate member fracture, the fracture criterion and fracture model of the FPM were developed. Furthermore, to simulate the rigidity of a steel beam–column connection, an independent zero-length element was adopted, which can consider the influence of nonlinear and hysteretic connection stiffness. The nonlinear response results were compared with those of existing studies to verify the accuracy of the proposed numerical procedure. Additionally, the collapse analysis of a Vogel six-story frame showed that frames with semi-rigid connections have greater anti-collapse capacity than those with rigid connections do.

회전 효과를 고려한 Active Gurney Flap 의 동특성 해석

Abstract : In this study, the finite element analysis was carried out to investigate dynamic characteristics of the AGF(Active Gurney Flap) which is under development for reducing vibration and noise of the helicopter rotor system. The Gurney flap is a kind of small flat plate, mounted normal to the lower surface of the airfoil near to the trailing edge. An electric motor, L-shaped linkages and flap parts were integrated into a rotor bade, and 3~5/rev control was given to the AGF to reduce the vibration in the fixed frame. Thus, an explicit time integration method was adopted to investigate the dynamic response of the AGF with considering both centrifugal force due to the rotor rotation and active control input, and it can be seen that the vertical displacement of the AGF was satisfied to meet the design requirement. § 이 논문은 대한기계학회 2014년도 추계학술대회(2014. 11. 11-14., 김대중컨벤션센터) 발표내용을 토대로 한 논문임. † Corresponding Author, naltlguy@kari.re.kr Ⓒ 2015 The Korean Society of Mechanical Engineers

A monotonicity preserving, nonlinear, finite element upwind method for the transport equation

Abstract We propose a simple upwind finite element method that is monotonicity preserving and weakly consistent of order O ( h 3 2 ) . The scheme is nonlinear, but since an explicit time integration method is used the added cost due to the nonlinearity is not prohibitive. We prove the monotonicity preserving property for the forward Euler method and for a second order Runge–Kutta method. The convergence properties of the Runge–Kutta finite element method are verified on a numerical example.

Role of Time Integration in Computing Transitional Flows Caused by Wall Excitation

Numerical investigation of receptivity and flow transition in spatio-temporal framework have shown the central role of spatio-temporal wave-front (STWF) created by wall excitation for transition of a two-dimensional (2D) zero pressure gradient boundary layer (ZPGBL) in Sengupta and Bhaumik (Phys Rev Lett 107:154501, 2011). Although the STWF is created by linear mechanism, it is the later nonlinear stage of evolution revealed by the solution of Navier---Stokes equation (NSE), which causes formation of turbulent spots merging together to create fully developed turbulent flow. Thus, computing STWF for ZPGBL from NSE is of prime importance, which has been reported by the present authors following earlier theoretical investigation. Similar computational efforts using NSE by other researchers do not report finding the STWF. In the present investigation we identify the main reason for other researchers to miss STWF, as due to taking a very short computational domain. Secondly, we show that even one takes a long enough domain and detect STWF, use of traditional low accuracy method will not produce the correct dynamics as reported by Sengupta and Bhaumik (2011). The role of time integration plays a very strong role in the dynamics of transitional flows. We have shown here that implicit methods are more error prone, as compared to explicit time integration methods during flow transition. For the present problem, it is noted that the classical Crank---Nicolson method is unstable for 2D NSE. Same error-prone nature will also be noted for hybrid implicit---explicit time integration methods (known as the IMEX methods). One of the main feature of present analysis is to highlight the accuracy of computations by compact schemes used by the present investigators over a significantly longer domain and over unlimited time, as opposed to those reported earlier in the literature for the wall excitation problem. A consequence of taking long streamwise domain enables one to detect special properties of STWF and its nonlinear growth. The main focus of the present research is to highlight the importance of STWF, which is a new class of spatio-temporal solution obtained from the linear receptivity by solving Orr---Sommerfeld equation and nonlinear analysis of NSE.

An explicit time-stepping technique for elastic waves under concepts of Green’s functions computed locally by the FEM

This work presents a new time-marching scheme able to reduce spurious (unphysical) oscillations by means of algorithm damping ratio for elastic wave propagation problems in the framework of the Explicit Green’s Approach (ExGA) (Mansur WJ, et al. J Comp Phys 2007; 227:851–870). The integral expression concerned with the ExGA is written in terms of the Green’s and the Step response functions. Their computations are carried out independently by means of the semidiscrete FEM and the Central difference method. Due to the principle of causality, the Green’s and Step response functions admit a compact support surround the source points for a small enough time step that is usually employed in common explicit time integration methods applied to wave propagation modeling. In this sense, the Green’s and Step response functions at t=Δt can be efficiently computed locally through small subdomains. Each local subdomain with its respective submesh covers only nodes whose Green’s and Step response function values do not vanish. The accuracy and effectiveness of the proposed methodology are demonstrated by analyzing three numerical examples.

A numerical study of the influence from pre‐existing cracks on granite rock fragmentation at percussive drilling

SummaryThe aim of this study is to investigate the effect of pre‐existing, or structural, cracks on dynamic fragmentation of granite. Because of the complex behavior of rock materials, a continuum approach is employed relying upon a plasticity model with yield surface locus as a quadratic function of the mean pressure in the principal stress space coupled with an anisotropic damage model. In particular, Bohus granite rock is investigated, and the material parameters are chosen based on previous experiments. The equation of motion is discretized using a finite element approach, and the explicit time integration method is employed. The pre‐existing cracks are introduced in the model by considering sets of elements with negligible tensile strength that leads to their immediate failure when loaded in tension even though they still carry compressive loads as crack closure occurs because of compressive stresses. Previously performed edge‐on impact tests are reconsidered here to validate the numerical model. Percussive drilling is simulated, and the influence of the presence of pre‐existing cracks is studied. The results from the analysis with different crack lengths and orientations are compared in terms of penetration stiffness and fracture pattern. It is shown that pre‐existing cracks in all investigated cases facilitate the drilling process. Copyright © 2014 John Wiley & Sons, Ltd.

A fully discrete stable discontinuous Galerkin method for the thin film epitaxy problem without slope selection

In this paper, we develop an energy stable fully discrete discontinuous Galerkin (DG) finite element method for the thin film epitaxy problem. Based on the method of lines, we construct and prove the energy stability of the spatial semi-discrete DG scheme firstly. To avoid the strict time step restriction of the explicit time integration method, the first order convex splitting method is used to get an unconditionally stable fully discrete DG method, which is a linearly implicit scheme for this nonlinear problem. The energy stability of the fully discrete convex splitting DG scheme is also proved. To improve the temporal accuracy, spectral deferred correction (SDC) method is adapted to achieve the high order accuracy in both time and space. Combining with the convex splitting method, the SDC method can be linearly solvable, high order accurate and stable in our numerical tests. These advantages ensure that the resulting fully discrete DG scheme is efficient to perform the long time simulation of the thin film epitaxy model. Numerical experiments of the accuracy and long time simulation show the capability and efficiency of the method.

Analysis of adaptive mesh refinement for IMEX discontinuous Galerkin solutions of the compressible Euler equations with application to atmospheric simulations

The resolutions of interests in atmospheric simulations require prohibitively large computational resources. Adaptive mesh refinement (AMR) tries to mitigate this problem by putting high resolution in crucial areas of the domain. We investigate the performance of a tree-based AMR algorithm for the high order discontinuous Galerkin method on quadrilateral grids with non-conforming elements. We perform a detailed analysis of the cost of AMR by comparing this to uniform reference simulations of two standard atmospheric test cases: density current and rising thermal bubble. The analysis shows up to 15 times speed-up of the AMR simulations with the cost of mesh adaptation below 1% of the total runtime. We pay particular attention to the implicit–explicit (IMEX) time integration methods and show that the ARK2 method is more robust with respect to dynamically adapting meshes than BDF2. Preliminary analysis of preconditioning reveals that it can be an important factor in the AMR overhead. The compiler optimizations provide significant runtime reduction and positively affect the effectiveness of AMR allowing for speed-ups greater than it would follow from the simple performance model.

Accurate solutions of wave propagation problems under impact loading by the standard, spectral and isogeometric high-order finite elements. Comparative study of accuracy of different space-discretization techniques

For the first time, accurate numerical solutions to impact problems have been obtained with the standard, spectral, and isogeometric high-order finite elements. Spurious high-frequency oscillations appearing in numerical results are quantified and filtered out by the two-stage time-integration approach. We also use the 1-D impact problem with a simple analytical solution for the comparison of accuracy of the different space-discretization techniques used for transient acoustics and elastodynamics problems. The numerical results show the computational efficiency of the linear finite elements with reduced dispersion compared with other space-discretization techniques used for elastodynamics with implicit and explicit time-integration methods. We also show that for all space-discretization methods considered (except the linear finite elements with the lumped mass matrix), very small time increments which are much smaller than stability limit should be used in basic computations at large observation times. We should note that the size of time increments used at the filtering stage and calculated according to the special formulas defines the range of actual frequencies and can be used as a quantitative measure for the comparison and prediction of the accuracy of different space-discretization techniques. We also show that the new findings are valid in the multidimensional case.

Accurate finite element modeling of acoustic waves

In the paper we suggest an accurate finite element approach for the modeling of acoustic waves under a suddenly applied load. We consider the standard linear elements and the linear elements with reduced dispersion for the space discretization as well as the explicit central-difference method for time integration. The analytical study of the numerical dispersion shows that the most accurate results can be obtained with the time increments close to the stability limit. However, even in this case and the use of the linear elements with reduced dispersion, mesh refinement leads to divergent numerical results for acoustic waves under a suddenly applied load. This is explained by large spurious high-frequency oscillations. For the quantification and the suppression of spurious oscillations, we have modified and applied a two-stage time-integration technique that includes the stage of basic computations and the filtering stage. This technique allows accurate convergent results at mesh refinement as well as significantly reduces the numerical anisotropy of solutions. We should mention that the approach suggested is very general and can be equally applied to any loading as well as for any space-discretization technique and any explicit or implicit time-integration method.

Stability criterion to explicit finite difference applied to the Bresse system

In this work, we show that the stability criterion of the explicit time integration method applied to the Bresse system is given by $$\begin{aligned} \Delta t\le \displaystyle \frac{2\epsilon }{\sqrt{ \bigg (12 +\displaystyle \frac{\epsilon ^2}{R^2}\bigg )}\displaystyle \frac{k G}{\rho }}, \end{aligned}$$ where the thickness \(\epsilon \) constitutes a limitation to compute the numerical solutions. This restriction to the stability criterion is not standard (is not CFL condition) and if \(\epsilon <<1\) it is very restrictive to numerical computations. To overcome this restriction, we use the technics performed by Wright [Commun Appl Numer Methods 3:181–185 (1987), Commun Numer Methods Eng 14:81–86 (1998)] to minimize the influence of \(\epsilon \) on stability criterion such that the CFL condition prevails.

A second-order accurate in time IMplicit–EXplicit (IMEX) integration scheme for sea ice dynamics

Current sea ice models use numerical schemes based on a splitting in time between the momentum and continuity equations. Because the ice strength is explicit when solving the momentum equation, this can create unrealistic ice stress gradients when using a large time step. As a consequence, noise develops in the numerical solution and these models can even become numerically unstable at high resolution. To resolve this issue, we have implemented an iterated IMplicit–EXplicit (IMEX) time integration method. This IMEX method was developed in the framework of an already implemented Jacobian-free Newton–Krylov solver. The basic idea of this IMEX approach is to move the explicit calculation of the sea ice thickness and concentration inside the Newton loop such that these tracers evolve during the implicit integration. To obtain second-order accuracy in time, we have also modified the explicit time integration to a second-order Runge–Kutta approach and by introducing a second-order backward difference method for the implicit integration of the momentum equation. These modifications to the code are minor and straightforward. By comparing results with a reference solution obtained with a very small time step, it is shown that the approximate solution is second-order accurate in time. The new method permits to obtain the same accuracy as the splitting in time but by using a time step that is 10 times larger. Results show that the second-order scheme is more than five times more computationally efficient than the splitting in time approach for an equivalent level of error.

Finite element modeling of linear elastodynamics problems with explicit time-integration methods and linear elements with the reduced dispersion error

We have developed two finite element techniques with reduced dispersion for linear elastodynamics that are used with explicit time-integration methods. These techniques are based on the modified integration rule for the mass and stiffness matrices and on the averaged mass matrix approaches that lead to the numerical dispersion reduction for linear finite elements. The analytical study of numerical dispersion for the new techniques is carried out in the 1-D, 2-D and 3-D cases. The numerical study of the efficiency of the dispersion reduction techniques includes the two-stage time-integration approach with the filtering stage (developed in our previous papers) that quantifies and removes spurious high-frequency oscillations from numerical results. We have found that in contrast to the standard linear elements with explicit time-integration methods and the lumped mass matrix, the finite element techniques with reduced dispersion yield more accurate results at small time increments (smaller than the stability limit) in the 2-D and 3-D cases. The recommendations for the selection of the size of time increments are suggested. The new approaches with reduced dispersion can be easily implemented into existing finite element codes and lead to significant reduction in computation time at the same accuracy compared with the standard finite element formulations.

An explicit time integration method for structural dynamics using septuple B-spline functions

In this paper, an explicit time integration method with three parameters is proposed for structural dynamics using periodic septuple B-spline interpolation polynomial functions. In this way, by use of septuple Bsplines, the authors have proceeded to solve the DE of motion governing a single DOF system, and later, the presented method has been generalized for a multiple DOF system. In the proposed method, a direct recursive formula for response of the system was formulated on the basis of septuple B-spline interpolation approximation. In terms of the specific requirements of this proposed method, two initialization approaches are given for initial calculation. One is called direct initialization, and the other is indirect initialization. The stability analysis of the proposed method illustrates that, by use of adjustable parameters, a highfrequency response can be damped out without inducing excessive algorithmic damping in important low frequency modes. The computational accuracy and efficiency of the proposed method is demonstrated with three numerical examples, and the results from the proposed method are compared with those from some of the existent numerical methods, such as the Newmark and Wilsonmethods. The compared results show that the proposed method has high accuracy with low time consumption. Copyright © 2013 John Wiley & Sons, Ltd.

Granite rock fragmentation at percussive drilling – experimental and numerical investigation

SUMMARYThe aim of this study is to numerically model the fracture system at percussive drilling. Because of the complex behavior of rock materials, a continuum approach is employed relying upon a plasticity model with yield surface locus as a quadratic function of the mean pressure in the principal stress space coupled with an anisotropic damage model. In particular, Bohus granite rock is investigated, and the material parameters are defined based on previous experiments. This includes different tests such as direct tension and compression, three‐point bending, and quasi‐oedometric tests to investigate the material behavior at both tension and confined compression stress states. The equation of motion is discretized using a finite element approach, and the explicit time integration method is employed. Edge‐on impact tests are performed, and the results are used to validate the numerical model. The percussive drilling problem is then modeled in 3D, and the bit‐rock interaction is considered using contact mechanics. The fracture mechanism in the rock and the bit penetration‐ resisting force response are realistically captured by the numerical model. Copyright © 2013 John Wiley &amp; Sons, Ltd.

CFD simulation of bubbling fluidized bidisperse mixtures: Effect of integration methods and restitution coefficient

In this work we simulated the dynamics of a fluidized bed of a binary solid mixture using the isothermal multi-fluid model of the commercial CFD code Fluent 12. We focused the attention on the role of both restitution coefficient and integration methods on the dynamics of the bed, adopting a mixture of solids assorted in size (with constant density) as case study. We employed two methods of spatial discretization: first-order upwind scheme (FUS) and second-order upwind scheme (SUS). We investigated implicit versus explicit time integration methods as well. The numerical diffusion introduced by the FUS resulted in a low bubble fraction in the bed, in turn reducing solid mixing rates. Simulations carried out adopting the SUS were characterized by a reasonable bubble fraction and associated solid mixing rate. The latter method successfully predicted the transition to fully mixed, uniform fluidization conditions. The value of the restitution coefficient did not significantly affect the results of the simulations, as bubble volume fraction and jetsam concentration profiles did not show any significant change within the interval of restitution coefficient values investigated, with the exception of the nearly ideal value of 0.99.

A small-scale decomposition for 3D boundary integral computations with surface tension

An efficient, non-stiff boundary integral method for the initial value problem for interfacial Darcy flow (which is a model of porous media flow) in three space dimensions is presented. We consider a ‘doubly-periodic’ interface separating two fluids, with surface tension present at the boundary. Surface tension introduces high order (i.e., high derivative) terms in the governing equation, and this imposes a severe stability constraint on explicit time-integration methods. Furthermore, the high order terms appear in a nonlocal operator, which makes it difficult to design an efficient implicit method. The stiffness is removed by developing a small-scale decomposition in the spirit of prior work in the two-dimensional problem by Hou, Lowengrub, and Shelley. In order to develop this small-scale decomposition, we formulate the problem using a generalized isothermal parameterization of the free surface. An additional difficulty is the efficient calculation of the Birkhoff–Rott integral for the velocity of the interface. We present a new algorithm, based on Ewald summation, to compute this in O(NlogN) operations, where N is the number of interface grid points. Our non-stiff method is expected to apply widely to problems for doubly-periodic interfacial flow with surface tension which have a boundary integral formulation.

A NEW EXPLICIT TIME INTEGRATION METHOD FOR STRUCTURAL DYNAMICS

A family of new explicit time-integration method is proposed herein, which inherits the numerical characteristics of any existing implicit Runge–Kutta algorithms for a linear conservative system. Based on an exact derivation of the increment of mechanical energy, the method proposed is demonstrated to be unconditionally stable. Also, the stability condition of the proposed method is derived when applied to solving a nonlinear system. The characteristics of the proposed method are investigated by observing the mechanical-energy time history of a nonlinear conservative system. The numerical results can be explained by the stability condition derived in the nonlinear regime. Finally, the computational accuracy and efficiency between the Newmark time integration method and the proposed explicit method are compared in solving the dynamic response of a couple of linear oscillators.